Italian citizenship is based on the jus sanguinis. Anyone that is of Italian descent can have his citizenship recognized.
The main qualifier is that your Italian ancestor must not have interrupted the chain. Before 1992, Italian citizens taking up another citizenship automatically lost Italian citizenship.
As an American, you qualify only if your ancestor’s son was born before your ancestor naturalized as a U.S. citizen.
The only major problem in the citizenship application arises when your ancestor documents contain discrepancies. The most common discrepancies are in names and dates. Many immigrants changed their names upon arriving in the United States.
In such cases, the Italian Consulate will require a formal Court order recognizing that your ancestor is one and the same person.
The procedure varies from State to State, although the vast majority of documents to be corrected are from the State of New York.
The Once and the same person Court order can be obtained through the filing of a formal lawsuit against the New York Department of Health, which is responsible for birth and death certificates.
As any other lawsuit, it will be a very complicated matter to handle, and you should seek the assistance of a qualified Italian lawyer.
Once you got your certificate amended, your case will be approved and you will be allowed to apply for an Italian passport.
It is important to note that U.S. law does not prohibit American citizens to hold dual citizenship.